In the city centre means you’re being specific about which city centre you’re in. I live in Boston and I went shopping in the city centre. In a city centre is more vague. You’re telling me it’s the centre of a city but I have no idea which city. I like to go shopping in a city centre rather than a town because there are more shops to chose from. Etc At the city centre would be more commonly used to say exactly where you are. Where are you right now? I’m at the city centre, doing some shopping. Downtown is an American word and as I’m from England, I will let other teachers answer that one!

To add to Simon's answer: in most cities and towns in the US "downtown" and "the city center" are interchangeable in meaning, but "downtown" is the more frequently used term.

Happy Mid-Autumn festival

It was my pleasure to help!